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Theorem r0cld 23462
Description: The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
r0cld ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ {𝑧 ∈ 𝑋 ∣ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)} ∈ (Clsdβ€˜π½))
Distinct variable groups:   π‘₯,π‘œ,𝑦,𝑧,𝐴   π‘œ,𝐽,π‘₯,𝑦,𝑧   π‘œ,𝐹,𝑧   π‘œ,𝑋,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem r0cld
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
21kqffn 23449 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 Fn 𝑋)
323ad2ant1 1133 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 Fn 𝑋)
4 fncnvima2 7062 . . . 4 (𝐹 Fn 𝑋 β†’ (◑𝐹 β€œ {(πΉβ€˜π΄)}) = {𝑧 ∈ 𝑋 ∣ (πΉβ€˜π‘§) ∈ {(πΉβ€˜π΄)}})
53, 4syl 17 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (◑𝐹 β€œ {(πΉβ€˜π΄)}) = {𝑧 ∈ 𝑋 ∣ (πΉβ€˜π‘§) ∈ {(πΉβ€˜π΄)}})
6 fvex 6904 . . . . . 6 (πΉβ€˜π‘§) ∈ V
76elsn 4643 . . . . 5 ((πΉβ€˜π‘§) ∈ {(πΉβ€˜π΄)} ↔ (πΉβ€˜π‘§) = (πΉβ€˜π΄))
8 simpl1 1191 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
9 simpr 485 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) β†’ 𝑧 ∈ 𝑋)
10 simpl3 1193 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
111kqfeq 23448 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) ↔ βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
12 eleq2w 2817 . . . . . . . . 9 (𝑦 = π‘œ β†’ (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ π‘œ))
13 eleq2w 2817 . . . . . . . . 9 (𝑦 = π‘œ β†’ (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ π‘œ))
1412, 13bibi12d 345 . . . . . . . 8 (𝑦 = π‘œ β†’ ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)))
1514cbvralvw 3234 . . . . . . 7 (βˆ€π‘¦ ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ))
1611, 15bitrdi 286 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)))
178, 9, 10, 16syl3anc 1371 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π΄) ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)))
187, 17bitrid 282 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) ∈ {(πΉβ€˜π΄)} ↔ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)))
1918rabbidva 3439 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ {𝑧 ∈ 𝑋 ∣ (πΉβ€˜π‘§) ∈ {(πΉβ€˜π΄)}} = {𝑧 ∈ 𝑋 ∣ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)})
205, 19eqtrd 2772 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (◑𝐹 β€œ {(πΉβ€˜π΄)}) = {𝑧 ∈ 𝑋 ∣ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)})
211kqid 23452 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐹 ∈ (𝐽 Cn (KQβ€˜π½)))
22213ad2ant1 1133 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (KQβ€˜π½)))
23 simp2 1137 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (KQβ€˜π½) ∈ Fre)
24 simp3 1138 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
25 fnfvelrn 7082 . . . . . 6 ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
263, 24, 25syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐹)
271kqtopon 23451 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
28273ad2ant1 1133 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
29 toponuni 22636 . . . . . 6 ((KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹) β†’ ran 𝐹 = βˆͺ (KQβ€˜π½))
3028, 29syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ ran 𝐹 = βˆͺ (KQβ€˜π½))
3126, 30eleqtrd 2835 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ βˆͺ (KQβ€˜π½))
32 eqid 2732 . . . . 5 βˆͺ (KQβ€˜π½) = βˆͺ (KQβ€˜π½)
3332t1sncld 23050 . . . 4 (((KQβ€˜π½) ∈ Fre ∧ (πΉβ€˜π΄) ∈ βˆͺ (KQβ€˜π½)) β†’ {(πΉβ€˜π΄)} ∈ (Clsdβ€˜(KQβ€˜π½)))
3423, 31, 33syl2anc 584 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ {(πΉβ€˜π΄)} ∈ (Clsdβ€˜(KQβ€˜π½)))
35 cnclima 22992 . . 3 ((𝐹 ∈ (𝐽 Cn (KQβ€˜π½)) ∧ {(πΉβ€˜π΄)} ∈ (Clsdβ€˜(KQβ€˜π½))) β†’ (◑𝐹 β€œ {(πΉβ€˜π΄)}) ∈ (Clsdβ€˜π½))
3622, 34, 35syl2anc 584 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ (◑𝐹 β€œ {(πΉβ€˜π΄)}) ∈ (Clsdβ€˜π½))
3720, 36eqeltrrd 2834 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (KQβ€˜π½) ∈ Fre ∧ 𝐴 ∈ 𝑋) β†’ {𝑧 ∈ 𝑋 ∣ βˆ€π‘œ ∈ 𝐽 (𝑧 ∈ π‘œ ↔ 𝐴 ∈ π‘œ)} ∈ (Clsdβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  {csn 4628  βˆͺ cuni 4908   ↦ cmpt 5231  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   Fn wfn 6538  β€˜cfv 6543  (class class class)co 7411  TopOnctopon 22632  Clsdccld 22740   Cn ccn 22948  Frect1 23031  KQckq 23417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-qtop 17457  df-top 22616  df-topon 22633  df-cld 22743  df-cn 22951  df-t1 23038  df-kq 23418
This theorem is referenced by:  nrmr0reg  23473
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